QUANTITATIVE FINANCE RESEARCH CENTRE. Regime Switching Rough Heston Model QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE

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1 QUANTITATIVE FINANCE RESEARCH CENTRE QUANTITATIVE F INANCE RESEARCH CENTRE QUANTITATIVE FINANCE RESEARCH CENTRE Research Paper 387 January 2018 Regime Switching Rough Heston Model Mesias Alfeus and Ludger Overbeck ISSN

2 Regime Switching Rough Heston Model Mesias Alfeus Ludger Overbeck December 12, 2017 Abstract We consider the implementation and pricing under a regime switching rough Heston model combining the approach by Elliott et al. (2016) with the one by Euch and Rosenbaum (2016). Key words: Rough Browian Motion, Regime Switching, Heston Model, Analytic Pricing Formula, Full and partial Monte-Carlo-Methods 1 Introduction The most celebrated and widely used stochastic volatility model is the model by Heston (1993). In that model the asset price S follows a geometric Brownian motion and the stochastic volatility follows a square-root-process, also known as CIR-process pionered by Cox et al. (1985). The dynamics of this model under the risk-neutral probability measure Q is given by: University of Technology Sydney, Australia, Mesias.Alfeus@student.uts.edu.au University of Giessen, Germany, Ludger.Overbeck@math.uni-giessen.de 1

3 ds t = rs t dt + S t Vt db t dv t = κ(θ V t )dt + σ V t dw t, (1) with two possible correlated Brownian motions B and W. One important advantage of this stochastic volatility model is its analytic tractability. It enables the modeller to infer the parameters of the process from the market quoted option prices. However, in a recent paper by Gatheral et al. (2014), it is shown that time series of realized volatility are rough with a Hurst parameter H less than onehalf, in particular near zero or of 0.1. In addition in Jaisson and Rosenbaum (2016) and Euch and Rosenbaum (2017) a micro-structure market model is based on self-exciting Poisson process, so called Hawkes processes, which converge to rough Brownian motions. In the rough Heston model the Brownian motion W driving the volatility is replaced by a rough (fractional-) Brownian motion W H, H (0, 0.5). Equation (1) can be re-written in a fractional stochastic volatility framework as follows: ds t = rs t dt + S t Vt db t dv t = κ(θ V t )dt + σ V t dw H t. (2) See also Alfeus et al. (2017) for numerical implementation of rough Heston model. Another stream of research, as described in Elliott et al. (2005) and Elliott et al. (2016) argues that asset prices or the associated volatility process should exhibit changing regime. They referred to example on the statistical analysis by Maghrebi et al. (2014), that the model should have at least two regimes under the risk neutral measure. Also several papers (Hamilton and Susmel, 2

4 1994; Moore and Wang, 2007; So et al., 1998) showed that index volatilities are subjected to regime switches under the physical measure. The economic consideration is one important motivation to use regime switches using Markov chains instead of jump-diffusions in order to incorporate sudden changes in volatility. Also a combination of rough Brownian motion and jump processes seems not to be considered in the literature as of yet. We restrict ourselves to changes in the mean-reversion parameter since that model maintains the analytic tractability. As a consequence many option pricing formulae can be obtain at least in a semi-analytic form. Another argument for regime switching models are those used for pricing in many other cases, like Overbeck and Weckend (2017), Yuen and Yang (2010), Alexander and Kaeck (2008), and Ang and Bekaert (2002). Calibration of the regime switching models have been analysed in Mitra (2009) and He and Zhu (2017). In the case of rough Heston model calibration is still a major open problem, since even employing semi-analytic solutions is a computationally expensive exercise. Since stochastic volatility models are usually not complete, there are several equivalent martingale measure. As long as volatility is not traded, the so-called minimal martingale measure will not change the volatility process and therefore regime-switching will be also passed on to the pricing measure. Our paper will now combine the two important generalization of the classical Heston model, namely the rough volatility model and regime switching volatilty. The so-called rough regime switching Heston model will inherit the analytic tractability of the rough Heston model, which was derived in Euch and Rosenbaum (2016, 2017) and the tractability of the regime switching extension as in Elliott et al. (2016). Two important stylized features of volatility, namely the rough behaviour in its local behaviour, and the regime switching property consistent with more long term economic consideration can be accommodated. In the classical Heston model the Laplace-transform of the log asset price 3

5 is a solution to a Riccati-equation. Although this result require the semimartingle and Markov-property of the asset and volatility process, a totally analogous result can be proved for the rough Heston model, where the volatility is neither a semi-martingle nor a Markov process. The Riccatti equation, which is an ordinary differenial equation is now replace by a rough integral equation, see Euch and Rosenbaum (2016). Moreover this results is extended to a time dependent long term mean reversion level θ s, s [0, T ]. Exactly in this formula time dependent θ is required in order to extend the resolvent equation as in Elliott et al. (2005) and Elliott et al. (2016) to our case. However, in our setting the resolvent equation, which is an equation associated with the Markovian regime switching process for θ s, depends also on the final time T. Based on the combination of the arguments from Euch and Rosenbaum (2016) and Elliott et al. (2016) we therefore derive an analytic representation of the Laplace-functional of the asset price. By standard Fourier-inversion technique analytic pricing formulae for put and calls are given. We benchmark these semi-analytic prices against two types of Monte- Carlo-simulations. One is a full Monte-Carlo simulation, in which the three dimensional stochastic processes (B, W, θ) is simulated and the option payout can be obtained (in the risk neutral world) in each simulation. The second is the partial Monte-Carlo. Here we only simulate the path of θ s (ω), s [0, T ] and then solve the corresponding rough Ricatti equation. Here we only avoid the resolvent equation which was shown to be very time consuming. The results are very close. This is in contrast to the results of Elliott et al. (2016). For reason not explained they only considered maturities upto year 1 and it is apparent that the difference between MC and analytic increases with maturity. This we can not observe. As a test we run the Monte-Carlo simulation without changing the regime, i.e. do not simulate the Markov chain of regime switches. Then our results are closer to the Monte-Carlo based figures reported by Elliott et al. (2016) in their numerical results. 4

6 In the section numerical results, we present the three different calculation methods. In addition we show that the call option price as a function of the Hurst parameter can exhibit different shapes. We see in our example that for shorter maturities call prices are increasing with increasing Hurst parameter, i.e. rough prices based on rough volatility are cheaper than those based Brownian motion prices and prices based on long memory volatility are even more expensive than Brownian motion. This changes if maturity increases. At a certain level Brownian volatility prices are the most expensive one and both rough and long term volatility based prices are lower (see Figure 1). We also analyse the sensitivity with respect to average number of regime until maturity, with respect to initial volatility and the correlation between W H and B. 2 Basic Model Description We directly work under the pricing measure for the underlying (already discounted) asset S. From Equation (2) the log prices X = log S then become dx t = (r V t /2)dt + V t db t dv t = κ(θ V t )dt + σ V t dw H t. (3) We now incorporate a regime switching into the mean reversion level θ as in Elliott et al. (2016) and choose the rough volatility model by Euch and Rosenbaum (2016). This leads to the following stochastic integral equation for V 5

7 V t V 0 = κ Γ(α) t + σ Γ(α) (t s) α 1 (θ s V s )dt 0 t 0 (t s) α 1 V s dw s, (4) where W is now a standard Brownian motion having correlation ρ with B and (θ s ) 0 t< is a finite state time homogeneous Markov process with generator matrix Q independent of S and W. 2.1 Fixed function s θ s We need the following result from Euch and Rosenbaum (2017) that for a fixed function s θ s the characteristic function of X t = log S t equals ( t E[e zxt ] = exp h(z, t s) (κθ s + V ) ) 0s α ds, z C, (5) 0 Γ(1 α) where h is the unique solution of the following fractional Riccati equation: D α h = 1 2 (z2 z) + (zρσ κ)h(z, s) + σ2 2 h2 (z, s), s < t, z C,(6) I 1 α h(z, 0) = 0. Here the fractional differentiation and integral are defined by D α h(z, s) = I α h(z, s) = t 1 (t s) α h(z, s)ds (7) Γ(1 α) 0 1 t (t s) α 1 h(z, s)ds (8) Γ(α) Regime switching θ s As in Elliott et al. (2016) we define θ s (ω) = k i=1 ϑ iz s (i) (ω) = ϑ, Z s where Z is a Markov chain, independent from (S, V ) with state space the set of unit 6

8 vectors in R k, i.e.z s {e i = (0,.., 1, 0..) T, i = 1,.., k} and ϑ is the vector of k-different mean reversion levels. The infinitesimal generator of the process Z is also denoted by Q i.e. q ij is the intensity of switching from state e i to e j, i.e. for θ itself the intensity of switching from ϑ i to ϑ j. Because of the independence we have E[e zx T ] = E [ ( T )] T exp κ h(z, T s) ϑ, Z s ds e 0 0 We fix the final time T and consider now the processes We have that ( g t = exp κ t 0 ) h(z, T s) ϑ, Z s ds h(z,t s) V 0 s α Γ(1 α) ds. (9) (10) G t = g t Z t (11) dg t = g t dz t + Z t dg t (12) = g t (Q Z t dt + dm Z t ) + Z t g t h(z, T t) ϑ, Z t dt and can proceed exactly as in Elliott et al. (2016). Therefore dg t = (Q + κh(z, t) ϑ, Z t ) g t Z t dt + g t dm Z t (13) = (Q + κh(z, T t)θ) g t Z t dt + g t dm Z t Once this is done we will finally end up with a matrix ODE as in Elliott et al. (2016), i.e. dφ(u, t) dt = (Q + κh(z, T t)θ) Φ(u, t), u < t, with Φ(u, u) = I.(14) 7

9 We now get that E [G t ] = Φ(0, t)z 0, (15) and because t, Z t, 1 = 1, we have E [ ( T )] exp κ h(z, s) ϑ, Z s ds = Φ(0, T )Z 0, 1. (16) 0 In summary, combining (8), (9), and (16) the regime switching rough Heston model has the characteristic representation given by: ϕ X (z) = E[e zx T ] = exp ( V 0 I 1 α h(z, T ) ) Φ(0, T )Z 0, 1. (17) This characteristic function is used in the semi-analytic pricing method below. 2.3 Monte-Carlo Simulation As benchmark for the semi-analytic pricing method based on the rough Ricatti equation and the matrix equation. We carry out two types of Monte- Carlo simulation. In the first one only the regime switching process is simulated and for each path of θ the corresponding Laplace-functional is calculated. In that way the performance of the ordinary matrix differential equation is tested against Monte-Carlo simulation. The second one is a straightforward simulation of the three dimensional process (θ, V, S). 8

10 2.3.1 Partial Monte-Carlo We only simulate the paths of θ s and then evaluate for each realization θ s (ω), the formula (5). A path θ(ω) has the form θ s (ω) = 1 [Si 1 (ω),s i (ω)[(s)x i (ω), (18) i=1 where S 0 = 0, S i (ω) = S i 1 (ω) + T i (ω), where T i, X i are successively drawn from an exponential distribution with parameter q Xi 1 (ω)x i 1 (ω) and X from the jump distribtuion of Q i.e. P [X i = θ k X i 1 ] = T i exp( q Xi 1 (ω)x i 1 (ω)) (19) q Xi 1 (ω)k q Xi 1 (ω)x i 1.(ω) (20) Let us generate N of those paths θ(ω l ), l = 1,.., N and evaluate for each θ(ω l ) the expression then ( t E[e zxt ](ω l ) := exp h(z, t s) (κθ s ((ω l )) + V ) ) 0s α ds, (21) 0 Γ(1 α) E[e zxt ] 1 N N E[e zxt ](ω l ) (22) l= Full Monte-Carlo Here we want to calculate the option price directly be Monte-Carlo simulation. We first simulate the 3-dimensional process (B, W H, θ). From the θ s (ω) simulated as above, we build the values of the regimes at each of the discrete time steps t i, at which we also want to generate the values of the volatility V ti, which depends on θ ti and the asset price S ti. The Heston model itself is then defined via an Euler scheme according 9

11 to (2), and option prices are obtained by evaluating the payoff at each path and taking the average over all MC-paths. 2.4 Analytic Pricing based on Fourier transformation To price options, we use the well-known Fourier-inversion formula of Gil- Pelaez (1951) (for convergence analysis see Wendel (1961)) which leads to a semi-analytic closed-form solution given by: C 0 = e rt E [ (e X K) +] = E[e X ]Π 1 e rt KΠ 2, (23) where the probability quantities Π 1 and Π 2 are given by: Π 1 = E[e X I {e X >K}]/E[e X ] = { } K iz ϕ X (z i) R dz π 0 izϕ X ( i) Π 2 = P{e X > K} = { } K iz ϕ X (z) R dz. π iz 0 (24) 3 Numerical results Most of the model parameters are adopted from Elliott et al. (2016), see Table 1. For the roughness case, we chose the Hurst parameter H = 0.1, as indicated by Gatheral et al. (2014) (see also Alfeus et al. (2017) for Hurst parameter estimation from realized variance). Our first anlysis begins with the test of the observation in Elliott et al. (2016) that with longer time to maturity Monte-Carlo prices diverge considerable from analytic prices. This we can not confirm. In percentage of price the Monte-Carlo error only increases slightly. These results are displayed in Table 2 3. However if we do not simulate the regime switches in the Monte Carlo simulation we observe the same increase as reported in Elliott et al. (2016), see Table 4. In our implementation we could neither observe the problems with maturity larger than 1 year nor the problems with 10

12 the discontinouities in the complex plane as reported in Elliott et al. (2016). However, we can only observe that semi-analytic pricing suffers for the out of the money options. In the second analysis we show how the Hurst parameter impacts the option price. Surprisingly this depends on the maturity of the option. We show the result without regime switching in the Figure 1. We get increasing, hump and then decreasing shapes for time to expiry bigger than 1.85 years. Thirdly, we report on call prices under rough volatility (with H = 0.1) with different volatility and correlation assumptions, see Tables 5 6. Here we also exhibit the partial Monte-Carlo results. The prices are close to full Monte-Carlo, but it is faster, and sometimes even closer to semi-analytic. Approximately simulations of the regime switches consume the same computation time as the semi-analytic calculations. Different to the Q E -matrix used by Elliott et al. (2016) which roughly allows for one change per year, we mainly consider the case with multiple switches per year. The differences to the non-regime switching becomes large, and these numerical results are given in Tables 7 8. Our last figure shows the implied volatility surface for different Hurst parameter, but with two different starting volatility but the same regime switching parameters and maturity T = 1. The lowest and steepest is the most rough one in the lowest regime, see Figure 2. This is naturally since option prices are cheaper under those parameters. 11

13 Table 1: Model parameters Parameters value S(0) 100 r 0.05 K 100 σ 0.4 ρ -0.5 κ 3 θ 0 = [θ 1 θ 2 ] [ ] α 1 [ ( H = 0.5) ] 1 1 Q E No. of Simulations Time Steps

14 Table 2: Call prices, v 0 = 0.02 < θ 1 < θ 2 (a) Starting in a low state: θ 0 = θ 1 K/T Price std Error Fourier Price std Error Fourier Price std Error Fourier (b) Starting in a high state: θ 0 = θ 2 K/T Price std Error Fourier Price std Error Fourier Price std Error Fourier

15 Table 3: Call prices, v 0 = 0.02 < θ 1 < θ 2, with longer maturities (a) Starting in a low state: θ 0 = θ 1 K/T Price std Error Fourier Price std Error Fourier Price std Error Fourier (b) Starting in a high state: θ 0 = θ 2 K/T Price std Error Fourier Price std Error Fourier Price std Error Fourier

16 Table 4: Call prices, v 0 = 0.02 < θ 1 < θ 2, without simulation of regime switches in Monte Carlo (a) Starting in a low state: θ 0 = θ 1 K/T Price std Error Fourier Price std Error Fourier Price std Error Fourier (b) Starting in a high state: θ 0 = θ 2 K/T Price std Error Fourier Price std Error Fourier Price std Error Fourier

17 Call option prices Call option prices Hurst parameter H (a) 6 months maturity Hurst parameter H (b) 1 year maturity Call option prices Call option prices Hurst parameter H (c) 1.85 years maturity Hurst parameter H (d) 2.5 years maturity Figure 1: The impact of Hurst parameter on option values with changing expiry time Under the rough case, we consider a case when H = 0.1 as empirically proven by Gatheral et al. (2014). At moment we are considering the generator matrix Q E given above. 16

18 Table 5: Call prices under rough volatility, v 0 = 0.02 < θ 1 < θ 2 and Q E generator (a) Starting in a low state: θ 0 = θ 1 K/T Full std Error Partial Fourier Full std Error Partial Fourier Full std Error Partial Fourier (b) Starting in a high state: θ 0 = θ 2 K/T Full std Error Partial Fourier Full std Error Partial Fourier Full std Error Partial Fourier In what follows, we consider a generator of the Markov chain Q = [ Meaning, we are considering 5 jump rate per year from state 1 to state 2 and 4 jump rate from state 2 to state 1. ]. 17

19 Table 6: Call prices under rough volatility and v 0 = 0.02 < θ 1 < θ 2 (a) Starting in a low state: θ 0 = θ 1 K/T Full std Error Partial Fourier Full std Error Partial Fourier Full std Error Partial Fourier (b) Stating in a high state: θ 0 = θ 2 K/T Full std Error Partial Fourier Full std Error Partial Fourier Full std Error Partial Fourier

20 Table 7: Call prices under regime-changing and rough volatility (a) Starting in a low state: θ 0 = θ 1 K/T Full Partial Fourier Full Partial Fourier Full Partial Fourier rho = 0, v0 = 0.05 ρ = 0.5, v0 = 0.1 ρ = 0.5, v0 = 0.1 (b) Starting in a high state: θ 0 = θ 2 K/T Full Partial Fourier Full Partial Fourier Full Partial Fourier ρ = 0, v 0 = 0.05 ρ = 0.5, v 0 = 0.1 ρ = 0.5, v 0 =

21 Table 8: Call prices under regime-changing and rough volatility (a) Starting in a low state: θ 0 = θ 1 K/T Full Partial Fourier Full Partial Fourier Full Partial Fourier ρ = 0.5, v 0 = 0.05 ρ = 0.5, v 0 = 0.05 ρ = 0.5, v 0 = 0.1 (b) Starting in a high state: θ 0 = θ 2 K/T Full Partial Fourier Full Partial Fourier Full Partial Fourier ρ = 0.5, v 0 = 0.05 ρ = 0.5, v 0 = 0.05 ρ = 0.5, v 0 =

22 Implied Volatility - Regime Switching Rough Heston Model H=0.1, low regime H=0.3, high regime H=0.5, high regime Implied Volatility Strike Maturity 3 Figure 2: Implied volatility surface for changing Hurst parameter H under different initial volatility regime 4 Conclusion In this paper we studied the regime switching rough Heston models. It combines the recently developed theory of rough volatility using rough Riccattiequations as in Euch and Rosenbaum (2016) and the regime swiching extension of the Heston model as in Elliott et al. (2016). Since there isn t yet any model combining rough Brownian motion with jumps and because of the analytic tractability we opt for the regime switching using hidden Markov chain instead of jumps. This enables us to incorporate sudden changes even in the rough volatility case. The call option price is still given in a semi-analytic 21

23 formula. We formulate and fully implemented this analytic approach to the rough switching Heston model, and implement as well two simulation based methods. The first is the full Monte-Carlo-Simulation of the underlying stochastic process and the second one is just the simulation of the regime switching Markov process, then applying the Ricatti equation and the Fourier methods for the call option. Our results show that the deviation between the approaches are small and consistent for any given time to expiry. We also analyse sensitivity to several input parameters. In particular, we show the sensitivity with respect to roughness, the Hurst parameter H. Actually this sensitivity depends on the time to expiry of the option. Concerning the regime switches we analyse Q-matrices, one with only one change per year as carried out in Elliott et al. (2016) and one with fast changes, approximately 5 per year. There is only a slight impact on call prices, see Table 5 and 6. 22

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